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      【NOI2019模拟赛（四十五）】bipartite
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        <h3 id="题意简述"><a class="markdownIt-Anchor" href="#题意简述"></a> 题意简述</h3>
<p>你有一个<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">n</span></span></span></span>个点的图，你需要进行<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi></mrow><annotation encoding="application/x-tex">m</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.43056em;"></span><span class="strut bottom" style="height:0.43056em;vertical-align:0em;"></span><span class="base textstyle uncramped"><span class="mord mathit">m</span></span></span></span>次操作</p>
<a id="more"></a>
<p>每次操作给一条边，若边已存在则删去这条边，否则加入这条边</p>
<p>每次操作后，判定这个图是否为二分图</p>
<h3 id="题解"><a class="markdownIt-Anchor" href="#题解"></a> 题解</h3>
<p>根据定义，若图中存在奇环，则不是二分图</p>
<p>于是不难想到维护一棵生成树，然后就可以方便地判定奇环</p>
<p>我们约定：连接树上两个深度奇偶性相同点的非树边构成的集合，为“奇边集合”</p>
<p>现在麻烦的是，如果删去了一条树边，就要考虑奇边集合中是否存在能顶上去的边</p>
<p>我们显然不希望这种情况发生。于是我们设一条边的边权为它删去的时间（钦定<span class="katex"><span class="katex-mathml"><math><semantics><mrow><mi>m</mi><mo>+</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">m+1</annotation></semantics></math></span><span class="katex-html" aria-hidden="true"><span class="strut" style="height:0.64444em;"></span><span class="strut bottom" style="height:0.72777em;vertical-align:-0.08333em;"></span><span class="base textstyle uncramped"><span class="mord mathit">m</span><span class="mbin">+</span><span class="mord mathrm">1</span></span></span></span>时刻将剩余的所有边删去），用LCT维护最大生成树</p>
<p>具体地，当加入一条边时，先把它所在的环拎出来，如果环中最小边比它小，显然要把小的边顶掉。于是将小的边cut掉，并加入奇边集合，将要加入的边link起来即可</p>
<p>化边为点是肯定要的，点的点权设为正无穷，然后维护权值最小的点编号，就可以支持上述操作了</p>
<p>这样，删去树边就可以直接cut，删去奇边集合中的边就直接删去</p>
<p>二分图的判定条件就是奇边集合是否为空</p>
<div class="highlight-box" autocomplete="off" autocorrect="off" autocapitalize="off" spellcheck="false" contenteditable="true" data-rel="CPP"><figure class="iseeu highlight /cpp"><table><tr><td class="gutter"><pre><span class="line">1</span><br><span class="line">2</span><br><span class="line">3</span><br><span class="line">4</span><br><span class="line">5</span><br><span class="line">6</span><br><span class="line">7</span><br><span class="line">8</span><br><span class="line">9</span><br><span class="line">10</span><br><span class="line">11</span><br><span class="line">12</span><br><span class="line">13</span><br><span class="line">14</span><br><span class="line">15</span><br><span class="line">16</span><br><span class="line">17</span><br><span class="line">18</span><br><span class="line">19</span><br><span class="line">20</span><br><span class="line">21</span><br><span class="line">22</span><br><span class="line">23</span><br><span class="line">24</span><br><span 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class="line">117</span><br></pre></td><td class="code"><pre><span class="line"><span class="meta">#<span class="meta-keyword">include</span><span class="meta-string">&lt;bits/stdc++.h&gt;</span></span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> ls ch[x][0]</span></span><br><span class="line"><span class="meta">#<span class="meta-keyword">define</span> rs ch[x][1]</span></span><br><span class="line"><span class="keyword">using</span> <span class="keyword">namespace</span> <span class="built_in">std</span>;</span><br><span class="line"> </span><br><span class="line"><span class="keyword">const</span> <span class="keyword">int</span> N=<span class="number">200010</span>;</span><br><span class="line"><span class="keyword">int</span> fa[N],ch[N][<span class="number">2</span>],flip[N];</span><br><span class="line"><span class="keyword">int</span> val[N],mnp[N],sz[N];</span><br><span class="line"><span class="keyword">int</span> stk[N],top=<span class="number">0</span>;</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">Edge</span>&#123;</span><span class="keyword">int</span> u,v,w;&#125; e[N];</span><br><span class="line"><span class="class"><span class="keyword">struct</span> <span class="title">OPT</span>&#123;</span><span class="keyword">int</span> id,ty;&#125; opt[N];</span><br><span class="line"><span class="built_in">map</span>&lt;<span class="keyword">int</span>,<span class="keyword">int</span>&gt; pre[N];</span><br><span class="line"><span class="keyword">int</span> n,m,odd=<span class="number">0</span>;</span><br><span class="line"><span class="keyword">bool</span> istree[N],isodd[N];</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">bool</span> <span class="title">nroot</span><span class="params">(<span class="keyword">int</span> y)</span></span>&#123;<span class="keyword">return</span> ch[fa[y]][<span class="number">0</span>]==y||ch[fa[y]][<span class="number">1</span>]==y;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">pushr</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;swap(ls,rs);flip[x]^=<span class="number">1</span>;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">pushdown</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(!flip[x]) <span class="keyword">return</span>;</span><br><span class="line">    <span class="keyword">if</span>(ls) pushr(ls);</span><br><span class="line">    <span class="keyword">if</span>(rs) pushr(rs);</span><br><span class="line">    flip[x]=<span class="number">0</span>;</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">maintain</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    mnp[x]=x;</span><br><span class="line">    <span class="keyword">if</span>(ls&amp;&amp;val[mnp[ls]]&lt;val[mnp[x]]) mnp[x]=mnp[ls];</span><br><span class="line">    <span class="keyword">if</span>(rs&amp;&amp;val[mnp[rs]]&lt;val[mnp[x]]) mnp[x]=mnp[rs];</span><br><span class="line">    sz[x]=sz[ls]+sz[rs]+(x&gt;n);</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">rotate</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> y=fa[x],z=fa[y];</span><br><span class="line">    <span class="keyword">int</span> k=(ch[y][<span class="number">1</span>]==x),w=ch[x][k^<span class="number">1</span>];</span><br><span class="line">    <span class="keyword">if</span>(nroot(y)) ch[z][ch[z][<span class="number">1</span>]==y]=x;</span><br><span class="line">    ch[x][k^<span class="number">1</span>]=y;ch[y][k]=w;</span><br><span class="line">    <span class="keyword">if</span>(w) fa[w]=y;</span><br><span class="line">    fa[x]=z;fa[y]=x;</span><br><span class="line">    maintain(y);</span><br><span class="line">    maintain(x);</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">splay</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> y=x,z;</span><br><span class="line">    stk[++top]=y;</span><br><span class="line">    <span class="keyword">while</span>(nroot(y)) y=fa[y],stk[++top]=y;</span><br><span class="line">    <span class="keyword">while</span>(top) pushdown(stk[top--]);</span><br><span class="line">    <span class="keyword">while</span>(nroot(x))</span><br><span class="line">    &#123;</span><br><span class="line">        y=fa[x];z=fa[y];</span><br><span class="line">        <span class="keyword">if</span>(nroot(y)) rotate((ch[y][<span class="number">0</span>]==x)^(ch[z][<span class="number">0</span>]==y)?x:y);</span><br><span class="line">        rotate(x);</span><br><span class="line">    &#125;</span><br><span class="line">    maintain(x);</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">access</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;<span class="keyword">for</span>(<span class="keyword">int</span> y=<span class="number">0</span>;x;y=x,x=fa[x]) splay(x),rs=y,maintain(x);&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">int</span> <span class="title">find</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;access(x);splay(x);<span class="keyword">while</span>(ls) pushdown(x),x=ls;<span class="keyword">return</span> x;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">makeroot</span><span class="params">(<span class="keyword">int</span> x)</span></span>&#123;access(x);splay(x);pushr(x);&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">split</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;makeroot(x);access(y);splay(y);&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">link</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;makeroot(x);fa[x]=y;&#125;</span><br><span class="line"><span class="function"><span class="keyword">inline</span> <span class="keyword">void</span> <span class="title">cut</span><span class="params">(<span class="keyword">int</span> x,<span class="keyword">int</span> y)</span></span>&#123;split(x,y);fa[x]=ch[y][<span class="number">0</span>]=<span class="number">0</span>;&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">insert</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">int</span> u=e[x].u,v=e[x].v,y;</span><br><span class="line">    <span class="keyword">if</span>(find(u)!=find(v)) istree[x]=<span class="number">1</span>,link(u,x+n),link(v,x+n);</span><br><span class="line">    <span class="keyword">else</span></span><br><span class="line">    &#123;</span><br><span class="line">        split(u,v);y=mnp[v]-n;</span><br><span class="line">        <span class="keyword">if</span>(e[x].w&gt;e[y].w)</span><br><span class="line">        &#123;</span><br><span class="line">            istree[x]=<span class="number">1</span>;istree[y]=<span class="number">0</span>;</span><br><span class="line">            <span class="keyword">if</span>(~sz[v]&amp;<span class="number">1</span>) odd++,isodd[y]=<span class="number">1</span>;</span><br><span class="line">            cut(e[y].u,y+n);cut(e[y].v,y+n);</span><br><span class="line">            link(u,x+n);link(v,x+n);</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span> <span class="keyword">if</span>(~sz[v]&amp;<span class="number">1</span>) odd++,isodd[x]=<span class="number">1</span>;</span><br><span class="line">    &#125;</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">void</span> <span class="title">remove</span><span class="params">(<span class="keyword">int</span> x)</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="keyword">if</span>(istree[x]) istree[x]=<span class="number">0</span>,cut(e[x].u,x+n),cut(e[x].v,x+n);</span><br><span class="line">    <span class="keyword">else</span> <span class="keyword">if</span>(isodd[x]) isodd[x]=<span class="number">0</span>,odd--;</span><br><span class="line">&#125;</span><br><span class="line"> </span><br><span class="line"><span class="function"><span class="keyword">int</span> <span class="title">main</span><span class="params">()</span></span></span><br><span class="line"><span class="function"></span>&#123;</span><br><span class="line">    <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&amp;n,&amp;m);</span><br><span class="line">    <span class="built_in">memset</span>(val,<span class="number">0x3f</span>,<span class="keyword">sizeof</span>(val));</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=m;i++) val[n+i]=m+<span class="number">1</span>;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>,u,v;i&lt;=m;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="built_in">scanf</span>(<span class="string">"%d%d"</span>,&amp;u,&amp;v);</span><br><span class="line">        <span class="keyword">if</span>(u&gt;v) swap(u,v);</span><br><span class="line">        <span class="keyword">if</span>(pre[u][v])</span><br><span class="line">        &#123;</span><br><span class="line">            e[pre[u][v]].w=i;</span><br><span class="line">            val[pre[u][v]+n]=i;</span><br><span class="line">            opt[i]=&#123;pre[u][v],<span class="number">2</span>&#125;;</span><br><span class="line">            pre[u][v]=<span class="number">0</span>;</span><br><span class="line">        &#125;</span><br><span class="line">        <span class="keyword">else</span> e[i]=&#123;u,v,m+<span class="number">1</span>&#125;,pre[u][v]=i,opt[i]=&#123;i,<span class="number">1</span>&#125;;</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">for</span>(<span class="keyword">int</span> i=<span class="number">1</span>;i&lt;=m;i++)</span><br><span class="line">    &#123;</span><br><span class="line">        <span class="keyword">if</span>(opt[i].ty==<span class="number">1</span>) insert(opt[i].id);</span><br><span class="line">        <span class="keyword">if</span>(opt[i].ty==<span class="number">2</span>) remove(opt[i].id);</span><br><span class="line">        <span class="built_in">puts</span>(odd?<span class="string">"NO"</span>:<span class="string">"YES"</span>);</span><br><span class="line">    &#125;</span><br><span class="line">    <span class="keyword">return</span> <span class="number">0</span>;</span><br><span class="line">&#125;</span><br></pre></td></tr></table></figure></div>
      
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